Finite Group Databasebeta

← #44402255

Subgroups — #44402255

Order 29056.

688 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20300273C211yesmaximal, characteristic
2C2×#20300250C211yesmaximal, characteristic
3C2×C42×D227C211yesmaximal, characteristic, complemented
4#28408924C211yesmaximal, characteristic, complemented
5#28408923C211yesmaximal, characteristic, complemented
6#28408913C211yesmaximal, characteristic, complemented
7#28408910C211yesmaximal, characteristic
8#28408906C211yesmaximal, characteristic, complemented
9#28408894C211yesmaximal, characteristic, complemented
10C4⋊C4×Dic227C211yesmaximal, characteristic, complemented
11#28408883C211yesmaximal, characteristic, complemented
12#28408879C211yesmaximal, characteristic, complemented
13#28408879C211yesmaximal, characteristic, complemented
14#28408879C211yesmaximal, characteristic, complemented
15#387×C227C211yesmaximal, characteristic, complemented
16C22×D908C2211yescharacteristic
17C22×C4×D227C2211yescharacteristic
18C22×C4×D227C2211yescharacteristic
19C22×C4×D227C2211yescharacteristic
20C22×C4×D227C2211yescharacteristic
21#20300273C418yescomplemented
22#20300250—24no—
23C2×C4⋊Dic227C2211yescharacteristic
24C2×Dic227⋊C4C2211yescharacteristic
25C2×Dic227⋊C4C2211yescharacteristic
26C2×(C2×D227)⋊C4C2211yescharacteristic
27C2×(C2×D227)⋊C4C2211yescharacteristic
28C2×(C2×D227)⋊C4C2211yescharacteristic
29C2×(C2×D227)⋊C4C2211yescharacteristic
30C2×(C2×D227)⋊C4C2211yescharacteristic
31C2×(C2×D227)⋊C4C2211yescharacteristic
32C2×(C2×D227)⋊C4C2211yescharacteristic
33C2×(C2×D227)⋊C4C2211yescharacteristic
34C2×C4×Dic227C2211yescharacteristic
35C2×C4×Dic227C2211yescharacteristic
36C2×C4×Dic227C2211yescharacteristic
37C2×C4×Dic227C2211yescharacteristic
38C2×C4×Dic227C2211yescharacteristic
39C2×C4×Dic227C2211yescharacteristic
40C42×D227—24no—
41C227⋊#95C2211yescharacteristic
42C227⋊#95C2211yescharacteristic
43C227⋊#95C2211yescharacteristic
44C227⋊#95C2211yescharacteristic
45C227⋊#95C2211yescharacteristic
46C227⋊#95C2211yescharacteristic
47C2×C4⋊C4×C227C2211yescharacteristic
48C2×C4⋊C4×C227C2211yescharacteristic
49C2×C42×C227C2211yescharacteristic
50#95×C227C2211yescharacteristic
51#95×C227C2211yescharacteristic
52#95×C227C2211yescharacteristic
53#95×C227C2211yescharacteristic
54C23×D227C2311yescharacteristic
55C23×D227C2311yescharacteristic
56C2×D908—22no—
57C2×D908—22no—
58C2×D908C2×C414yes—
59C22×Dic227C2311yescharacteristic
60C22×Dic227C2311yescharacteristic
61C22×Dic227C2311yescharacteristic
62C22×Dic227C2311yescharacteristic
63C22×Dic227C2311yescharacteristic
64C22×Dic227C2311yescharacteristic
65C2×C4×D227—22no—
66C2×C4×D227—22no—
67C2×C4×D227—22no—
68C2×C4×D227—22no—
69C2×C4×D227—22no—
70C2×C4×D227—22no—
71C2×C4×D227—22no—
72C2×C4×D227—22no—
73C2×C4×D227—22no—
74C2×C4×D227—22no—
75C2×C4×D227C2×C414yescomplemented
76C2×C4×D227D414yes—
77C4⋊Dic227—22no—
78Dic227⋊C4—22no—
79Dic227⋊C4C2×C414yescomplemented
80(C2×D227)⋊C4—22no—
81(C2×D227)⋊C4—22no—
82(C2×D227)⋊C4—22no—
83(C2×D227)⋊C4—22no—
84(C2×D227)⋊C4—22no—
85(C2×D227)⋊C4C2×C414yescomplemented
86(C2×D227)⋊C4C2×C414yescomplemented
87(C2×D227)⋊C4C2×C414yescomplemented
88C4×Dic227—22no—
89C4×Dic227—22no—
90C4×Dic227—22no—
91C4×Dic227—22no—
92C4×Dic227—22no—
93C4×Dic227—22no—
94C22×C4×C227C2311yescharacteristic
95C22×C4×C227C2311yescharacteristic
96C22×C4×C227C2311yescharacteristic
97C22×C4×C227C2311yescharacteristic
98C22×C4×C227C2311yescharacteristic
99C22×C4×C227C2311yescharacteristic
100C22×C4×C227C2311yescharacteristic
101C4⋊C4×C227—22no—
102C4⋊C4×C227C2×C414yescomplemented
103C42×C227—22no—
104C22×D227—21no—
105C22×D227—21no—
106C22×D227—21no—
107C22×D227—21no—
108C22×D227—21no—
109C22×D227—21no—
110C22×D227—21no—
111C22×D227—21no—
112C22×D227—21no—
113C22×D227—21no—
114C22×D227C22×C412yes—
115C22×D227C22×C412yes—
116C22×D227C2×D412yes—
117C22×D227#4112yes—
118D908—28no—
119C2×Dic227—21no—
120C2×Dic227—21no—
121C2×Dic227—21no—
122C2×Dic227—21no—
123C2×Dic227—21no—
124C2×Dic227—21no—
125C2×Dic227—21no—
126C2×Dic227—21no—
127C2×Dic227—21no—
128C2×Dic227—21no—
129C2×Dic227—21no—
130C2×Dic227—21no—
131C2×Dic227C22×C412yes—
132C2×Dic227C22×C412yes—
133C2×Dic227C2×D412yes—
134C2×Dic227#4112yes—
135C2×Dic227#4112yes—
136C2×Dic227#4112yes—
137C4×D227—28no—
138C4×D227—28no—
139C4×D227—28no—
140C4×D227—44no—
141C23×C227C2411yescharacteristic
142C2×C4×C227—21no—
143C2×C4×C227—21no—
144C2×C4×C227—21no—
145C2×C4×C227—21no—
146C2×C4×C227—21no—
147C2×C4×C227—21no—
148C2×C4×C227—21no—
149C2×C4×C227—21no—
150C2×C4×C227—21no—
151C2×C4×C227—21no—
152C2×C4×C227—21no—
153C2×C4×C227—21no—
154C2×C4×C227—21no—
155C2×C4×C227—21no—
156C2×C4×C227—21no—
157C2×C4×C227—21no—
158C2×C4×C227C22×C412yes—
159C2×C4×C227C22×C412yes—
160C2×C4×C227C22×C412yes—
161C2×C4×C227C2×D412yes—
162C2×C4×C227#4112yes—
163C22×C227C23×C411yescharacteristic
164C22×C227C22×D411yescharacteristic
165C22×C227C2×#4111yescharacteristic
166C22×C227C2×#4111yescharacteristic
167C22×C227C2×#4111yescharacteristic
168C22×C227C2×#4111yescharacteristic
169C22×C227C2×#4111yescharacteristic
170C2×D227—22no—
171C2×D227—22no—
172C2×D227—22no—
173C2×D227—22no—
174C2×D227—22no—
175C2×D227—22no—
176C2×D227—22no—
177C2×D227—22no—
178C2×D227—22no—
179C2×D227—41no—
180C2×D227—41no—
181C2×D227—41no—
182C2×D227—41no—
183C2×D227C4×D414yes—
184C4×C227—22no—
185C4×C227—22no—
186C4×C227—22no—
187C4×C227—22no—
188C4×C227—22no—
189C4×C227—41no—
190C4×C227—41no—
191Dic227—22no—
192Dic227—22no—
193Dic227—22no—
194Dic227—41no—
195Dic227—41no—
196Dic227#12314yescomplemented
197C2×C227C2×C4×D411yescharacteristic
198C2×C227C4×#4111yescharacteristic
199C2×C227C4×#4111yescharacteristic
200C2×C227C2×#12311yescharacteristic
201C2×C227#52511yescharacteristic
202C2×C227#52911yescharacteristic
203C2×C227#52911yescharacteristic
204D227—24no—
205D227—42no—
206C227C4×#12311yescharacteristic, complemented, Sylow
207C4×#123—2271nomaximal, Sylow
208C2×#123—2271no—
209C2×C4×D4—2271no—
210C22×C42—2271no—
211#388—2271no—
212#388—2271no—
213#387—2271no—
214#387D22711yescharacteristic, complemented
215#386—2271no—
216#385—2271no—
217#381—2271no—
218C4×C4⋊C4—2271no—
219C4×C4⋊C4—2271no—
220C4×C22⋊C4—2271no—
221C4×C22⋊C4—2271no—
222C4×C22⋊C4—2271no—
223C22×D4—2271no—
224C23×C4—2271no—
225C23×C4—2271no—
226C23×C4—2271no—
227C23×C4—2271no—
228#123—2278no—
229C4×D4—4544no—
230C2×C4⋊C4—2271no—
231C2×C4⋊C4—2271no—
232C2×C4⋊C4—2271no—
233C2×C4⋊C4C2×D22711yescharacteristic
234C2×C4⋊C4C2×D22711yescharacteristic
235C2×C22⋊C4—2271no—
236C2×C22⋊C4—2271no—
237C2×C22⋊C4—2271no—
238C2×C22⋊C4—2271no—
239C2×C22⋊C4—2271no—
240C2×C22⋊C4—2271no—
241C2×C22⋊C4—2271no—
242C2×C22⋊C4—2271no—
243C2×C42—4544no—
244C2×C42—2271no—
245C2×C42—2271no—
246C2×C42—2271no—
247C2×C42—2271no—
248C2×C42—2271no—
249C2×C42—2271no—
250C2×C42C2×D22711yescharacteristic
251#95—2271no—
252#95—2271no—
253#95—2271no—
254#95—2271no—
255#95—2271no—
256#95—2271no—
257#95C2×D22711yescharacteristic
258#95C2×D22711yescharacteristic
259#95C2×D22711yescharacteristic
260#95C2×D22711yescharacteristic
261C24—2271no—
262C24—2271no—
263C2×D4—4542no—
264C2×D4—4542no—
265C2×D4—2274no—
266C22×C4—4542no—
267C22×C4—4542no—
268C22×C4—4542no—
269C22×C4—4542no—
270C22×C4—4542no—
271C22×C4—4542no—
272C22×C4—4542no—
273C22×C4—4542no—
274C22×C4—4542no—
275C22×C4—4542no—
276C22×C4—2271no—
277C22×C4—2271no—
278C22×C4—2271no—
279C22×C4—2271no—
280C22×C4—2271no—
281C22×C4—2274no—
282C22×C4—2274no—
283C22×C4—2271no—
284C22×C4C22×D22711yescharacteristic
285C22×C4C22×D22711yescharacteristic
286C22×C4C22×D22711yescharacteristic
287C22×C4C22×D22711yescharacteristic
288C22×C4C22×D22711yescharacteristic
289C22×C4C22×D22711yescharacteristic
290C22×C4C22×D22711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—4542no—
293C4⋊C4—4542no—
294C4⋊C4—2274no—
295C4⋊C4C4×D22714yescomplemented
296C22⋊C4—4542no—
297C22⋊C4—4542no—
298C22⋊C4—4542no—
299C22⋊C4—4542no—
300C22⋊C4—4542no—
301C22⋊C4—2274no—
302C22⋊C4—2274no—
303C22⋊C4—2274no—
304C42—22no—
305C42—4542no—
306C42—4542no—
307C42—4542no—
308C42—4542no—
309C42—4542no—
310C42—4542no—
311C23—4541no—
312C23—4541no—
313C23—4541no—
314C23—4541no—
315C23—4541no—
316C23—4541no—
317C23—4541no—
318C23—4541no—
319C23—4541no—
320C23—4541no—
321C23—2272no—
322C23—2272no—
323C23—2272no—
324C23—2272no—
325C23C23×D22711yescharacteristic
326D4—4548no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—9084no—
344C2×C4—4548no—
345C2×C4—4548no—
346C2×C4—4548no—
347C2×C4—4541no—
348C2×C4—4541no—
349C2×C4—4541no—
350C2×C4—4541no—
351C2×C4—4541no—
352C2×C4—4541no—
353C2×C4—4541no—
354C2×C4—4541no—
355C2×C4—4541no—
356C2×C4—4541no—
357C2×C4—4541no—
358C2×C4—4541no—
359C2×C4—2272no—
360C2×C4—2272no—
361C2×C4—2272no—
362C2×C4—2272no—
363C2×C4—2272no—
364C2×C4—2272no—
365C2×C4C2×C4×D22712yes—
366C2×C4C2×C4×D22712yes—
367C2×C4C2×C4×D22712yes—
368C2×C4D4×D22712yes—
369C2×C4#1681388512yes—
370C22—9081no—
371C22—9081no—
372C22—9081no—
373C22—9081no—
374C22—4542no—
375C22—4542no—
376C22—4542no—
377C22—4542no—
378C22—4542no—
379C22—4542no—
380C22—4542no—
381C22—4542no—
382C22—4542no—
383C22—2274no—
384C22C22×C4×D22711yescharacteristic
385C22C2×D4×D22711yescharacteristic
386C22#41×D22711yescharacteristic
387C22#41×D22711yescharacteristic
388C22#41×D22711yescharacteristic
389C22#41×D22711yescharacteristic
390C22C2×#1681388511yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—41no—
397C4—41no—
398C4—9081no—
399C4—9081no—
400C4—4542no—
401C4—4542no—
402C4—4542no—
403C4—2274no—
404C2—9082no—
405C2—4544no—
406C2C4×D4×D22711yescharacteristic
407C2#2840950011yescharacteristic
408C2C4×#1681388511yescharacteristic
409C2#123×D22711yescharacteristic
410C2#2840954511yescharacteristic
411C2#2840956511yescharacteristic
412C2#2840956711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutatorC22×C227C23×C4
Fitting#387×C227C2
socle, cosocleC23×C227C24
prime radicalC22×D908C22
hypercenter#387D227
hypocenterC227C4×#123
LCSG ▷ C22×C227 ▷ C227
UCSC1 ◁ C23 ◁ #387

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×C42×D227C2semidirect
4. #28408924C2semidirect
5. #28408923C2semidirect
6. #28408913C2semidirect
8. #28408906C2semidirect
9. #28408894C2semidirect
10. C4⋊C4×Dic227C2semidirect
11. #28408883C2semidirect
12. #28408879C2semidirect
13. #28408879C2semidirect
14. #28408879C2semidirect
15. #387×C227C2semidirect
21. #20300273C4semidirect
75. C2×C4×D227C2×C4semidirect
79. Dic227⋊C4C2×C4semidirect
85. (C2×D227)⋊C4C2×C4semidirect
86. (C2×D227)⋊C4C2×C4semidirect
87. (C2×D227)⋊C4C2×C4semidirect
102. C4⋊C4×C227C2×C4semidirect
196. Dic227#123semidirect
206. C227C4×#123semidirect
214. #387D227semidirect
295. C4⋊C4C4×D227semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#203002732
C2×#203002502
C2×C42×D2272
#284089242
#284089232
#284089132
#284089102
#284089062
#284088942
C4⋊C4×Dic2272
#284088832
#284088792
#387×C2272
C4×#123227